Calculations of the Effect of Varying Hoop Dimensions
1st Draft, 27th February, 2009
The Nottingham and Hurlingham clubs are looking to get new hoops cast, with larger dimensions than standard Jaques hoops, to make them firmer, particularly for Championship play. Both clubs have alluvial soils, which tend to offer less resistance to hoop movement than is experienced in dry conditions elsewhere. This paper reports calculations that have been made in an attempt to assess the effect of increasing the diameter of the uprights and length and/or diameter of the carrots.
It has been reported that the WCF are changing their hoop specification to allow 3/4” uprights (instead of the usual 5/8”) and that the CA specification may follow suit. The possibility of carrots with a different vertical profile than the standard cone, as suggested by Alan Pidcock, has also been investigated, though investigations have been confined to circular cross-sections, rather than the fins being used in America.
A simplified model of a hoop was used, consisting of a single carrot and upright of cast iron, with a specific gravity of 7.1. A standard Jaques hoop was taken as the reference. Although at first glance it might be assumed to have conical carrots, the ones examined did not in practice narrow uniformly to a point, but are instead were truncated somewhat above where the point of the cone would have been and was roughly hemispherical below. They have been treated as truncated cones, with a circumference at the top of the carrot of 4.5 inches, and at the bottom of 1.5 inches, with a length of 8.25 inches. The carrot and upright were assumed to be rigid, any movement resulting from movement of the carrot in the soil.
Two different types of calculations were performed. The first was of the static resistance of the hoop to small horizontal displacements at half-ball and crown height, assuming that it was set in soil with a uniform depth profile. The force exerted by the soil on a horizontal slice of the carrot was assumed to be linearly proportional to the thickness of the slice, its diameter and the horizontal displacement. In other words, the soil was treated as a spring with a stiffness linearly dependent on the cross section of the carrot perpendicular to the direction of its displacement, and any effects of friction were ignored. The constant of proportionality was not determined, but is assumed to be a characteristic of the soil conditions and thus both location and weather dependent. The horizontal force on the upright at the height of interest and the position of the resulting centre of rotation, were then found by integrating this force and its moment along the length of the carrot and solving the two simultaneous equations for translational and rotational equilibrium.
The second was of the dynamics of an elastic collision between a ball, represented as a 1lb point mass traveling with some initial velocity perpendicular to the upright and hitting it at either half-ball or crown height, with the upright and carrot entirely unconstrained. The mass, position of the centre of mass and moment of inertia of the hoop/carrot were calculated from its geometry and density. Equations representing the conservation of linear momentum, angular momentum and energy were then solved to determine the final velocity of the ball (and that of the centre of mass of the hoop and its angular velocity).
Clearly, a fuller calculation should be possible to combine these effects, but it is hoped that investigating them separately will give enough insight into the effect of hoop geometry on a ball. The calculations were done using an algebraic manipulation package called “reduce”.
Upright is the diameter of the upright, Length the length of the carrot and Max Diameter the diameter of the top of the carrot (all in inches). The Mass is that calculated, in pounds using a specific gravity of 7.1, for a single hoop upright and carrot: it needs to be doubled and an allowance made for the crown to estimate the weight of an actual hoop. Vbf/Vbi is ratio of initial to final velocity of a ball hitting the unconstrained single upright/carrot: a negative value indicates that the ball rebouds; a positive one that it continues to travel forward. HB is for an impact at half-ball height (1.8125 inches); C for crown height (12 inches).
Truncated Cone (diameter of base =0.48 inches):
Fb is the ratio of the force resisting movement by a small displacement to the extent of that displacement and a constant characteristic to the soil conditions.
Truncated Cone (base diameter=0.48 inches):
The main conclusion I draw from the dynamic calculations is obvious one that increasing the mass of the hoop will cause a ball to bounce back faster from it (or, in the case of an impact at crown height, slow it down more). A secondary effect is that the higher the centre of mass the more effective it is: thus
Similarly, in the static calculations, the bigger the carrot the more effective it is in resisting movement (if the assumptions underlying the model are valid). However, the secondary effect is the other way round, particularly for impacts at crown height. For a pure cone, the force for a given displacement at half-ball height is proportional to:
d * l3 / (l2 + 7.25 * l + 19.7)
where d is the diameter of the top of the carrot and l its length.
This increases with the cube of the length for very short carrots, less than say 1 inch in length, with its square at about 4 inches and then tends gradually to a linear dependency as the length increases.
At crown height, the equivalent expression (which has the same constant of proportionality) is:
d * l3 / (l2 + 48 * l + 864)
This has the same behavior, but the curve is shifted so that dependence on length is much higher for plausible carrot lengths.
As to the idea of having a more complex profile than a truncated cone, the results suggest that any improvement over a pure cone of the same length and maximum diameter is likely to be marginal, to the extent of vanishing if the resistance offered by the vertical part of the profile is ignored (as it will be impossible to firm that up by knocking the carrot further in).
The overall conclusion from these calculations is that increasing the size and hence mass of the hoop should give a theoretical improvement in its resistance to an impact by a ball, but it is difficult to estimate the practical effect. For a given percentage increase in linear dimension, increasing the size of the upright has most effect on the dynamic properties, and increasing the length of the carrot for static ones. However, significantly increasing the length, but not the diameter of the top, of the carrots will mean that they need to be knocked in further to achieve a given amount of tightening when firming them up after they have become loosened by play (or taking them in and out of the ground). How far to increase the size is ultimately a matter of judgment, as it is a trade-off between the anticipated improvement in playing characteristics against the practicalities of setting, moving and storing larger hoops, particularly for clubs that have to bring them in at night.
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